UNIVERSITÀ DEGLI STUDI DI SALERNO Bachelor Degree in Chemical Engineering Course: Process Instrumentation and Control (Strumentazione e Controllo dei Processi Chimici) REFERENCE LINEAR DYNAMIC SYSTEMS First-Order Systems Rev. 2.5 – May 17, 2019

FIRST-ORDER LAG I.C.: t=0 y(0)=0 see: Ch.10 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice” I.C.: t=0 y(0)=0 First-order ODE, linear, non-homogeneous, with constant coefficients Forcing function: f(t) If a0≠0 CANONICAL FORM in the time domain in the Laplace domain 12/09/2019 Process Instrumentation and Control - Prof. M. Miccio

FIRST-ORDER LAG position of the pole in the complex plane Im Re -1/p Characteristic polynomial: tps + 1 Characteristic eq. : tps + 1 = 0 Only 1 pole: p = -1/tp < 0 NOTE: Self-regulating dynamic behavior 12/09/2019 Process Instrumentation and Control - Prof. M. Miccio

DIMENSIONLESS RESPONSE OF FIRST-ORDER LAG TO STEP INPUT CHANGE Forcing funtion: f(t)=Au(t) A=const. > 0 F(s)=A/s p t see: Ch.10 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice” NOTE: Self-regulating dynamic behavior 12/09/2019 Process Instrumentation and Control - Prof. M. Miccio

FIRST-ORDER PROCESSES Characteristic Parameters Static Gain ( Kp ): This is the ultimate value of the response (new steady-state) for a unit-step change in the input. ( Theorem of the final value) Time Constant (t p): The time constant of a process is a measure of the time necessary for the process to adjust to a change in its input. from: Romagnoli & Palazoglu (2005), “Introduction to Process Control” 12/09/2019 Process Instrumentation and Control - Prof. M. Miccio

FIRST-ORDER PROCESSES Effect of tp time 5 10 15 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 h(t), m 20 Response of the level of a tank (first-order system) to a unit-step change in the input flow rate from: Romagnoli & Palazoglu (2005), “Introduction to Process Control” 12/09/2019 Process Instrumentation and Control - Prof. M. Miccio

FIRST-ORDER PROCESSES Effect of tp The system eventually reaches a new equilibrium point (new steady-state). Two tanks with different cross sectional areas respond with different speeds (different time constants). The tank with a small area responds faster (smaller time constant). The tank with a larger area responds slower (larger time constant). from: Romagnoli & Palazoglu (2005), “Introduction to Process Control” 12/09/2019 Process Instrumentation and Control - Prof. M. Miccio

DIMENSIONLESS RESPONSE OF FIRST-ORDER LAG TO STEP INPUT CHANGE Dimensionless diagram of the dynamic response see: Ch.10 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice” NOTE: Self-regulating dynamic behavior 12/09/2019 Process Instrumentation and Control - Prof. M. Miccio

DIMENSIONLESS RESPONSE OF FIRST-ORDER LAG TO UNIT IMPULSE Forcing function : f(t)=(t) F(s) = 1 Dynamic response: NOTE: Self-regulating dynamic behavior 12/09/2019 Process Instrumentation and Control - Prof. M. Miccio

DIMENSIONLESS RESPONSE OF FIRST-ORDER LAG TO UNIT IMPULSE Dimensionless diagram of the dynamic response see: Ch.10 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice” NOTE: Self-regulating dynamic behavior 12/09/2019 Process Instrumentation and Control - Prof. M. Miccio

DIMENSIONLESS RESPONSE OF FIRST-ORDER LAG TO SINUSOIDAL INPUT Forcing function: Dynamic response for long time: - - - -f(t)  taken from: SCPC2-Modelli-rif_UniPI_2007-08.pdf Frequency  unchanged Amplitude ratio AR: B/A < 1 Phase lag <0 12/09/2019 Process Instrumentation and Control - Prof. M. Miccio

RESPONSE TO A SINUSOIDAL INPUT Homework: Diagram the DYNAMIC RESPONSE with the mod. Custom Process of LOOP-PRO Control Station 12/09/2019 Process Instrumentation and Control - Prof. M. Miccio

PURE CAPACITIVE SYSTEM or FIRST-ORDER INTEGRATOR If ao=0 CANONICAL FORM in the time domain in the Laplace domain 12/09/2019 Process Instrumentation and Control - Prof. M. Miccio

Process Instrumentation and Control - Prof. M. Miccio PURE CAPACITIVE FIRST-ORDER SYSTEM position of the pole on the complex plane Im Re Only 1 pole at the origin of the axes: s = 0 Dynamic response to step input change NOTE: marginally stable dynamic system  NON-self-regulating for some inputs 12/09/2019 Process Instrumentation and Control - Prof. M. Miccio

THE OPEN TANK WITH A VARIABLE LEVEL AS A FIRST-ORDER SYSTEM 1st-order after linearization b) pure capacitive see: Ch.10 - Stephanopoulos, "Chemical process control: an Introduction to theory and practice" 12/09/2019 Process Instrumentation and Control - Prof. M. Miccio

Process Instrumentation and Control - Prof. M. Miccio NONLINEAR FIRST-ORDER SYSTEMS 1st case: nonlinearity of the function of y(t) CANONICAL FORM where F(y(t)) is a nonlinear function of y(t) LINEARIZATION METHOD Taylor expansion at the first term: Where : ε=o(y-y0)2 o indicates the “order of magnitude” y0= ys= steady-state value of y(t) or y0=0 y0= another point of interest of y(t) 12/09/2019 Process Instrumentation and Control - Prof. M. Miccio

LINEARIZATION METHOD APPLICATION TO THE TANK WITH A VARIABLE LEVEL Mass balance equations in dynamic condition: at the steady-state: (1) I. C. : t=0 ; h(0) = hs (1 bis) R F0 Linearization by Taylor: Replacing in (1) : (2) At the steady state (h=hs; Accumulation=0): Thus, we have: (3) 12/09/2019 Process Instrumentation and Control - Prof. M. Miccio

LINEARIZATION METHOD APPLICATION TO THE TANK WITH A VARIABLE LEVEL Substracting eq. (3) from eq. (2) the deviation variable appears: I. C. : t=0 ; h’(0) = 0 Applying Laplace transform: L(I) = L(II) where: NOTE: The evaluated TF is valid only around the steady state. 12/09/2019 Process Instrumentation and Control - Prof. M. Miccio

LINEAR APPROXIMATION OF A NONLINEAR FUNCTION Due to the tangent theorem, the simulation of the linear approximation is satisfactory only around the the point x0 (hs) of linearization (it is not possible to extend the linearized function to the whole curve). Another linearization will be necessary to analyze the dynamic response of the system in another point. t0 t see: Ch.6 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice” t1 12/09/2019 Process Instrumentation and Control - Prof. M. Miccio

Process Instrumentation and Control - Prof. M. Miccio ORIGINAL MODEL and LINEARIZED MODEL: COMPARISON BETWEEN THE DYNAMIC RESPONSES APPLICATION TO THE TANK WITH A VARIABLE LEVEL Initial Condition Theoretical response see: Ch.6 - Stephanopoulos, "Chemical process control: an Introduction to theory and practice" 12/09/2019 Process Instrumentation and Control - Prof. M. Miccio

Process Instrumentation and Control - Prof. M. Miccio NONLINEAR FIRST-ORDER SYSTEM 2nd case: nonlinearity in the derivative term d [y(t)]/d t ORIGINAL FORM: where is a nonlinear function of y(t) e.g., LINEARIZATION METHOD  The linearization method is still applicable, with the Taylor expansion at the first term, but consequent elaboration and calculations necessary to obtain the mathematical model in terms of deviation variables and the TF are more difficult and not treated here! 12/09/2019 12/09/2019 Process Instrumentation and Control - Prof. M. Miccio 21 21